1,560 research outputs found
A Unifying Theory for Graph Transformation
The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Quantum ergodicity on the Bruhat-Tits building for in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
is a sequence of cocompact torsionfree lattices. We prove a form of
equidistribution on average for eigenfunctions whose spectral parameters lie in
the tempered spectrum when the associated sequence of quotients of the
Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most
fundamental techniques used in parameterized complexity. Unfortunately, even if
we consider concepts as simple as path or tree decompositions, such dynamic
programming uses space that is exponential in the decomposition's width, and
there are good reasons to believe that this is necessary. However, it has been
shown that in graphs of low treedepth it is possible to design algorithms which
achieve polynomial space complexity without requiring worse time complexity
than their counterparts working on tree decompositions of bounded width. Here,
treedepth is a graph parameter that, intuitively speaking, takes into account
both the depth and the width of a tree decomposition of the graph, rather than
the width alone.
Motivated by the above, we consider graphs that admit clique expressions with
bounded depth and label count, or equivalently, graphs of low shrubdepth (sd).
Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a
bounded-depth analogue of treewidth. We show that also in this setting,
bounding the depth of the decomposition is a deciding factor for improving the
space complexity. Precisely, we prove that on -vertex graphs equipped with a
tree-model (a decomposition notion underlying sd) of depth and using
labels, we can solve
- Independent Set in time using
space;
- Max Cut in time using space; and
- Dominating Set in time using space via
a randomized algorithm.
We also establish a lower bound, conditional on a certain assumption about
the complexity of Longest Common Subsequence, which shows that at least in the
case of IS the exponent of the parametric factor in the time complexity has to
grow with if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^(dk) ⋅ n^(1) using (dk²log n) space; - Max Cut can be solved in time n^(dk) using (dk log n) space; and - Dominating Set can be solved in time 2^(dk) ⋅ n^(1) using n^(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial
Colouring Complete Multipartite and Kneser-type Digraphs
The dichromatic number of a digraph is the smallest such that can
be partitioned into acyclic subdigraphs, and the dichromatic number of an
undirected graph is the maximum dichromatic number over all its orientations.
Extending a well-known result of Lov\'{a}sz, we show that the dichromatic
number of the Kneser graph is and that the
dichromatic number of the Borsuk graph is if is large
enough. We then study the list version of the dichromatic number. We show that,
for any and , the list
dichromatic number of is . This extends a recent
result of Bulankina and Kupavskii on the list chromatic number of ,
where the same behaviour was observed. We also show that for any ,
and , the list dichromatic number of the complete
-partite graph with vertices in each part is , extending
a classical result of Alon. Finally, we give a directed analogue of Sabidussi's
theorem on the chromatic number of graph products.Comment: 15 page
On the sizes of generalized cactus graphs
A cactus is a connected graph in which each edge is contained in at most one
cycle. We generalize the concept of cactus graphs, i.e., a -cactus is a
connected graph in which each edge is contained in at most cycles where
. It is well known that every cactus with vertices has at most
edges. Inspired by it, we attempt to
establish analogous upper bounds for general -cactus graphs. In this paper,
we first characterize -cactus graphs for based on the block
decompositions. Subsequently, we give tight upper bounds on their sizes.
Moreover, the corresponding extremal graphs are also characterized. However,
the case of remains open. For the case of 2-connectedness, the range
of is expanded to all positive integers in our research. We prove that
every -connected -cactus graphs with vertices has at most
edges, and the bound is tight if . But, for ,
determining best bounds remains a mystery except for some small values of .Comment: 14 pages, 2 figure
Structured Semidefinite Programming for Recovering Structured Preconditioners
We develop a general framework for finding approximately-optimal
preconditioners for solving linear systems. Leveraging this framework we obtain
improved runtimes for fundamental preconditioning and linear system solving
problems including the following. We give an algorithm which, given positive
definite with
nonzero entries, computes an -optimal
diagonal preconditioner in time , where is the
optimal condition number of the rescaled matrix. We give an algorithm which,
given that is either the pseudoinverse
of a graph Laplacian matrix or a constant spectral approximation of one, solves
linear systems in in time. Our diagonal
preconditioning results improve state-of-the-art runtimes of
attained by general-purpose semidefinite programming, and our solvers improve
state-of-the-art runtimes of where is the
current matrix multiplication constant. We attain our results via new
algorithms for a class of semidefinite programs (SDPs) we call
matrix-dictionary approximation SDPs, which we leverage to solve an associated
problem we call matrix-dictionary recovery.Comment: Merge of arXiv:1812.06295 and arXiv:2008.0172
The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees
We consider locally checkable labeling LCL problems in the LOCAL model of
distributed computing. Since 2016, there has been a substantial body of work
examining the possible complexities of LCL problems. For example, it has been
established that there are no LCL problems exhibiting deterministic
complexities falling between and . This line of
inquiry has yielded a wealth of algorithmic techniques and insights that are
useful for algorithm designers.
While the complexity landscape of LCL problems on general graphs, trees, and
paths is now well understood, graph classes beyond these three cases remain
largely unexplored. Indeed, recent research trends have shifted towards a
fine-grained study of special instances within the domains of paths and trees.
In this paper, we generalize the line of research on characterizing the
complexity landscape of LCL problems to a much broader range of graph classes.
We propose a conjecture that characterizes the complexity landscape of LCL
problems for an arbitrary class of graphs that is closed under minors, and we
prove a part of the conjecture.
Some highlights of our findings are as follows.
1. We establish a simple characterization of the minor-closed graph classes
sharing the same deterministic complexity landscape as paths, where ,
, and are the only possible complexity classes.
2. It is natural to conjecture that any minor-closed graph class shares the
same complexity landscape as trees if and only if the graph class has bounded
treewidth and unbounded pathwidth. We prove the "only if" part of the
conjecture.
3. In addition to the well-known complexity landscapes for paths, trees, and
general graphs, there are infinitely many different complexity landscapes among
minor-closed graph classes
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